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LCM of 32 and 56

LCM of 32 and 56 is the smallest number among all common multiples of 32 and 56. The first few multiples of 32 and 56 are (32, 64, 96, 128, . . . ) and (56, 112, 168, 224, 280, 336, . . . ) respectively. There are 3 commonly used methods to find LCM of 32 and 56 - by division method, by prime factorization, and by listing multiples.

1. LCM of 32 and 56
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 32 and 56?

Answer: LCM of 32 and 56 is 224.

LCM of 32 and 56

Explanation:

The LCM of two non-zero integers, x(32) and y(56), is the smallest positive integer m(224) that is divisible by both x(32) and y(56) without any remainder.

Methods to Find LCM of 32 and 56

The methods to find the LCM of 32 and 56 are explained below.

  • By Division Method
  • By Prime Factorization Method
  • By Listing Multiples

LCM of 32 and 56 by Division Method

LCM of 32 and 56 by Division Method

To calculate the LCM of 32 and 56 by the division method, we will divide the numbers(32, 56) by their prime factors (preferably common). The product of these divisors gives the LCM of 32 and 56.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 32 and 56. Write this prime number(2) on the left of the given numbers(32 and 56), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (32, 56) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
  • Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 32 and 56 is the product of all prime numbers on the left, i.e. LCM(32, 56) by division method = 2 × 2 × 2 × 2 × 2 × 7 = 224.

LCM of 32 and 56 by Prime Factorization

Prime factorization of 32 and 56 is (2 × 2 × 2 × 2 × 2) = 25 and (2 × 2 × 2 × 7) = 23 × 71 respectively. LCM of 32 and 56 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 25 × 71 = 224.
Hence, the LCM of 32 and 56 by prime factorization is 224.

LCM of 32 and 56 by Listing Multiples

To calculate the LCM of 32 and 56 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 32 (32, 64, 96, 128, . . . ) and 56 (56, 112, 168, 224, 280, 336, . . . . )
  • Step 2: The common multiples from the multiples of 32 and 56 are 224, 448, . . .
  • Step 3: The smallest common multiple of 32 and 56 is 224.

∴ The least common multiple of 32 and 56 = 224.

☛ Also Check:

LCM of 32 and 56 Examples

  1. Example 1: The GCD and LCM of two numbers are 8 and 224 respectively. If one number is 56, find the other number.

    Solution:

    Let the other number be a.
    ∵ GCD × LCM = 56 × a
    ⇒ a = (GCD × LCM)/56
    ⇒ a = (8 × 224)/56
    ⇒ a = 32
    Therefore, the other number is 32.

  2. Example 2: Verify the relationship between GCF and LCM of 32 and 56.

    Solution:

    The relation between GCF and LCM of 32 and 56 is given as,
    LCM(32, 56) × GCF(32, 56) = Product of 32, 56
    Prime factorization of 32 and 56 is given as, 32 = (2 × 2 × 2 × 2 × 2) = 25 and 56 = (2 × 2 × 2 × 7) = 23 × 71
    LCM(32, 56) = 224
    GCF(32, 56) = 8
    LHS = LCM(32, 56) × GCF(32, 56) = 224 × 8 = 1792
    RHS = Product of 32, 56 = 32 × 56 = 1792
    ⇒ LHS = RHS = 1792
    Hence, verified.

  3. Example 3: The product of two numbers is 1792. If their GCD is 8, what is their LCM?

    Solution:

    Given: GCD = 8
    product of numbers = 1792
    ∵ LCM × GCD = product of numbers
    ⇒ LCM = Product/GCD = 1792/8
    Therefore, the LCM is 224.
    The probable combination for the given case is LCM(32, 56) = 224.

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FAQs on LCM of 32 and 56

What is the LCM of 32 and 56?

The LCM of 32 and 56 is 224. To find the least common multiple of 32 and 56, we need to find the multiples of 32 and 56 (multiples of 32 = 32, 64, 96, 128 . . . . 224; multiples of 56 = 56, 112, 168, 224) and choose the smallest multiple that is exactly divisible by 32 and 56, i.e., 224.

What is the Least Perfect Square Divisible by 32 and 56?

The least number divisible by 32 and 56 = LCM(32, 56)
LCM of 32 and 56 = 2 × 2 × 2 × 2 × 2 × 7 [Incomplete pair(s): 2, 7]
⇒ Least perfect square divisible by each 32 and 56 = LCM(32, 56) × 2 × 7 = 3136 [Square root of 3136 = √3136 = ±56]
Therefore, 3136 is the required number.

How to Find the LCM of 32 and 56 by Prime Factorization?

To find the LCM of 32 and 56 using prime factorization, we will find the prime factors, (32 = 2 × 2 × 2 × 2 × 2) and (56 = 2 × 2 × 2 × 7). LCM of 32 and 56 is the product of prime factors raised to their respective highest exponent among the numbers 32 and 56.
⇒ LCM of 32, 56 = 25 × 71 = 224.

What are the Methods to Find LCM of 32 and 56?

The commonly used methods to find the LCM of 32 and 56 are:

  • Listing Multiples
  • Prime Factorization Method
  • Division Method

If the LCM of 56 and 32 is 224, Find its GCF.

LCM(56, 32) × GCF(56, 32) = 56 × 32
Since the LCM of 56 and 32 = 224
⇒ 224 × GCF(56, 32) = 1792
Therefore, the GCF (greatest common factor) = 1792/224 = 8.

 
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